Table Of ContentNon-Markovian Toy Quantum Chain
Andr´as Bodor,1 Lajos Di´osi,2 Zs´ofia Kallus,1 and Thomas Konrad3
1Department of Physics of Complex Systems, Eo¨tvo¨s University
H-1117 Budapest, Pa´zm´any P´eter s´et´any 1/A, Hungary
2Research Institute for Particle and Nuclear Physics
H-1525 Budapest 114, POB 49, Hungary
3School of Chemistry and Physics, University of KwaZulu-Natal
Private Bag X54001, Durban 4000, South Africa
(Dated: January 18, 2013)
Weproposeasimplestructureforstationarynon-Markovianquantumchainsintheframeworkof
collisional dynamics of open quantum systems. To this end, we modify the microscopic Markovian
system–reservoir model, consider multiple collisions with each of the molecules with an overlap
between the collisional time intervals. We show how the equivalent Markovian quantum chain
3 can be constructed with the addition of satellite quantum memory to the system. We distinguish
1 quantum from classical non-Markovianity. Moreover, we define the counts of non-Markovianity
0 by the required number of satellite qubits and bits, respectively. As the particular measure of
2 quantum non-Markovianity the discord of the satellite w.r.t. the system is suggested. Simplest
n qubit realizations are discussed, and the significance for real system–environment dynamics is also
a pointed out.
J
7
1 I. INTRODUCTION
]
h In recent years we have seen increasing interest in abstract modeling of time evolution of open quantum systems
p coupled to their environment via different quantum channels. While decrypting fundamental laws is their ultimate
- goal, inspirations of such models and possible future applications of their results range from solid state physics to
t
n quantum biology and quantum information technology [1–4].
a Dynamics modeled microscopically as a series of discrete interactions or “collisions” between a central system and
u
environmental(reservoir)molecules leadstoabstractcollisionalsystem-reservoirmodels. ForamemorylessMarkovian
q
time evolution the reservoir has the idealized ability of forgetting new information faster than the rate of collisions;
[
i.e., after each collision the reservoir would totally relax to its pre-collisional state before the next collision occurs.
1 For the (central) system, this results in what we call Markovian quantum chain. In each step along the chain, the
v
system undergoes decoherence, leakage of information out to the reservoir to be forgotten there. Mathematically,
2
the Markov quantum chain’s irreversible dynamics is obtained from reduction of the unitary time evolution of the
1
system-environment compound by means of partial trace over degrees of freedom of the environment. In the special
2
4 caseofstationarity,themathematicaltoolofsemigroupsofcompletelypositive(CP)tracepreservingdynamicalmaps
. can be used [2–6].
1
The collisional model of Markovian chains has been studied in numerous works [7, 8]. It will be particularly useful
0
if we want to monitor the system [9–11]. Although we are never allowed to directly measure the system state, we can
3
1 measure each reservoir molecule after its collision with the system. By their regular selective measurements we can
: monitor the system state indirectly. The corresponding resolved evolution of the quantum chain is called selective
v
Markovian quantum chain.
i
X A more realistic description requires an account of the reservoir memory and the induced non-Markovianity of the
r dynamics of the central system (the chain). For analyzing new dynamical properties and quantifying the difference
a
from the simple Markovian processes, several non-Markovianity measures have been proposed so far. These make use
ofdifferentaspectsofnon-Markovian(NM)evolutions,e.g.,thenon-divisibilityoftheunderlyingquantumdynamical
map [12], the increasing trace distance (i.e.: distinguishability) of two initial quantum states, accompanied by the
back-flow of information from the reservoir to the system [13], or the discord between system and reservoir [14].
Non-Markovianity is a field of active debates (cf., e.g., [15]) so we emphasize the importance of abstract models to
capture fundamental aspects of memory keeping processes.
In this paper we study the structural features of NM open quantum systems by constructing discretized NM
processes, i.e., quantum chains. Starting from the generic structure of a Markovian quantum chain we are going to
impose a certain NM structure on it. The proposed abstract NM structure corresponds to an open quantum system
in a reservoir of non-interacting molecules which only collide with the central system through unitary collisions. The
initial, for the moment Markovian, quantum chain can be seen in Fig. 1(a). To engineer the NM element, we allow
each molecule to collide with the system more than a single time, say twice, while the molecules’ collisional intervals
overlap,byassumption,withthecollisionalperiodsofbothpreviousandconsecutivemolecules,cf. Fig.1(b). Instead
of separating what is forgotten and what is not by different timescales, our abstract model has a built-in exact
2
ρ
U U ξ U U ξ U
ρ
ξ ξ U U U
ξ
(a) (b)
ρ
U
ξ U
ρ
ξ
U U U
ξ
ξ
(c) (d)
FIG. 1. Quantum chains: different system-reservoir collision models ((cid:37): system state, ξ: reservoir molecule state, U system-
molecule unitary collision operator). (a) Standard Markovian chain: the system interacts with each independent molecule
once. (b)TheproposedNMchain: thesysteminteractswitheachindependentmoleculetwice,withoverlapbetweencollisional
intervalsof’nearest’molecules. (c)NMchainin[18]: thesysteminteractswitheachmoleculeonlyonce,butthemoleculesare
initially entangled. (d) NM chain in [19]: the system interacts with the same molecule multiple times.
memory time. Most importantly, we show how our NM chains become Markovian at the price of attaching satellite
memory subsystems. DuetoourproposedspecificNMstructure,theidentificationofthesatellitememorywithinthe
environmentisstraightforwardcomparedespeciallytotheeffortsrequestedinoscillatoryreservoirmodels(cf. [16,17]
and Refs. therein). Then a natural count of non-Markovianity follows: let it be the size (in qubits) of the requested
satellite memory. We can refine this count into informatic measures of non-Markovianity, as we show later.
The NM quantum collisional dynamics is not new in itself. Alternative NM structures are shown in Figs. 1(c) and
1(d). Ref.[18]introducesNMmechanismbystartingallmoleculesfromanapriorientangledstate,cf. Fig.1(c). This
work focuses on the molecular realization of a single NM step instead of general NM chains. In [19], the environment
consists of a single molecule and this molecule collides with the system consecutively, many times, cf. Fig. 1(d). An
important advantage of our NM structure over these two is that it can invariably host the monitoring of the system,
whichwassoinstrumentalforquantumMarkovchainsbutbecomescomplicatedinstructureFig.1(c)andimpossible
in Fig. 1(d).
In Sec. II the exact notion of a non-selective and a selective quantum Markovian chain is defined. The definition is
extended for quantum NM chains in Sec. III. Then Sec. IV contains our new results demonstrated on the toy models
of one-qubit NM chains. Based on these results we discuss in Sec. V a new measure of quantum non-Markovianity as
well as a distinction between classical and truly quantum non-Markovianity.
II. QUANTUM MARKOV CHAIN
To define Markov chains, we apply the toolbox containing CP-maps, Kraus matrices, POVMs, selective and non-
selective quantum measurements, detailed by the monographs [2–6]. We understand by a non-selective Markov chain
a series of quantum states (cid:37) ,(cid:37) ,...,(cid:37) ... of a given system where each state along the chain depends but on the
0 1 t
preceeding state respectively:
(cid:37) =M (cid:37) ; t=0,1,2,..., (1)
t+1 t t
where M are CP maps. If not stated otherwise, we restrict ourselves to stationary chains M =M. A CP map can
t t
always be represented by certain Kraus matrices M such that
λ
(cid:88)
M(cid:37)= M (cid:37)M†. (2)
λ λ
λ
3
ρ
U U
ξ ξ
λ λ
FIG.2. SelectiveMarkovchain: therefinementofthenon-selectiveoneinFig.1(a). Thepost-collisionalstateofthemolecule
is measured, and yields the random outcome λ.
Accordingly, non-selective Markov chains (1) can be decomposed into selective Markov chains given by the recursion
relation:
1
(cid:37) = M (cid:37) M†. (3)
t+1 p λ t λ
λ,t
λ may represent the random outcome of a POVM measurement characterized by the matrices M . Here TrM (cid:37) M†
λ λ t λ
is the outcome probability. As a matter of fact, the selective chain (cid:37) depends on all measurement outcomes λ prior
t
to t, this dependence is suppressed in our notation. The stochastic average of the selective Markov chain (3) over the
measurement outcomes λ yields the non-selective Markov chain (1).
We can always construct a microscopic model for a given quantum Markov chain. Consider an abstract ideal gas
(reservoir) of identical molecules each in state ξ. Independent unitary collisions will generate the CP map M of (1):
(cid:37) =Tr [U((cid:37) ⊗ξ)U†], (4)
t+1 res t
where U is the collision matrix, Tr is the partial trace over the molecule state. The process is shown in Fig. 1(a).
res
Any map (2) can be generated by a suitable unitary mechanism (4) whereas the choice of U and ξ is never unique
[5, 6]. If we inspect, i.e. measure, the post-collisional state of the molecule we get the underlying microscopic model
of the selective Markov chain (3), shown in Fig. 2. As to determining Kraus matrices M , their possible choice is
λ
simple if the molecules are prepared in pure state ξ =|Ψ(cid:105)(cid:104)Ψ|:
M =(cid:104)λ|U|Ψ(cid:105). (5)
λ
We show concrete examples in Sec. IV.
III. QUANTUM NON-MARKOV CHAIN
Consider the chain
(cid:37) =M(t)(cid:37) ; t=0,1,2,..., (6)
t 0
where M(t) is a t-dependent CP map. The chain is Markovian, see (1), if M(t) is divisible, i.e. can be written in the
following form:
M(t)=M M(t−1); t=1,2,..., (7)
t
with some sequence of CP maps M ,M ,...,M ,.... To be clear, we require that
1 2 t
M(1)=M ,
1
M(2)=M M , (8)
2 1
M(3)=M M M , e.t.c.
3 2 1
If such a sequence M ,M ,...,M ,... does not exist, i.e., when M(t) is not divisible into the same set of factors
1 2 t
for all t, the chain (cid:37) is NM.
t
The microscopic mechanism of quantum non-Markovianity can be quite complicated. A recent study [18] considers
single qubit maps which are not divisible at all. To model such maps microscopically, a class of finite NM chains has
been constructed. Its key mechanism has been independent collisions with entangled molecules, the general structure
is shown in Fig. 1(c). Another work [19] considered a single qubit environment, interacting unlimited times with
the system qubit via a controlled-rotation. Our approach will be different and elementary. We construct a class of
4
λ
ξ ξ
ρ
ξ
λ λ
FIG. 3. Selective NM chain: the refinement of the non-selective one in Fig. 1(b). The post-collisional (i.e.: after the double
collision) state of the molecule is measured, and yields the random outcome λ.
ρ
ξ ξ
λ λ
FIG. 4. Selective one-qubit Markov chain: collisional model. The system qubit (cid:37) interacts (collides) with the reservoir qubits
(molecules) ξ via XOR-gates; the post-interaction states of the molecular qubits are measured, yielding the random outcomes
λ.
stationary non-Markovianity. Our molecules are independent, and do not become entangled prior to their collisions
with the system. However, they will collide multiple times — twice, for simplicity’s sake — and this constitutes the
memorymechanism. Fig.1(b)showsthegeneralNMstructure. UnlikethepreviousNMstructureshowninFig.1(d),
ours allows us, by its construction, to insert measurements without changing the physics of the process. Obviously,
we can insert measurements on each molecule after its second collision, as shown in Fig. 3.
Rather than discussing this NM model generally, in what follows we concentrate on its simplest qubit realizations.
IV. ONE-QUBIT NON-MARKOV CHAINS
Inordertohighlightthesimilaritiesanddifferences,first,inSec.IVA,wearegoingtoconstructaone-qubitMarkov
chain, and then, in Secs. IVB and IVC, by extending it, we study two different NM structures.
Throughoutthissectionthemoleculesconstitutingthereservoiraresinglequbitsinpureinitialstatesξ =|Ψ (cid:105)(cid:104)Ψ |,
φ φ
where
|Ψ (cid:105)=cosφ|0(cid:105)+sinφ|1(cid:105). (9)
φ
For later convenience, we write the state of the molecule as
|Ψ (cid:105)=exp(iφσ )|0(cid:105). (10)
φ y
A. One-qubit Markov chain
Let our central system be a single qubit, and let us construct a Markov chain (1-3), also see Figs. 1(a) and 2.
Couple the molecule in state ξ to the system qubit in state (cid:37) via the XOR-gate. The corresponding two-qubit
t
unitary operator U is the following:
U =XOR . (11)
sys−mol
The XOR-gate installs the unsharp measurement of the system qubit by the molecular qubit. The strength of the
measurementiscontrolledbytheparameterφofthemolecularpre-collisionalstate. Ifφ=0,π/2,themeasurementis
projectivewhileφ≈π/4yieldsaweakmeasurement[20];thevalueφ=π/4decouplesthesystemfromthemolecules.
The two Kraus matrices are defined by M =(cid:104)λ|U|Ψ (cid:105) for λ=0,1, cf. Eq. (5). Inserting (10), we have
λ φ
M =(cid:104)λ|Uexp(iφσ )|0(cid:105). (12)
λ y
5
To perform the partial trace operation (4) from the four-dimensional state U((cid:37) ⊗ξ)U† to the two-dimensional state
t
(cid:37) , we use the deeper level selective form (3) for technical reasons.
t
Theunitaryevolutionofthemolecule-systemcompoundξ⊗(cid:37) intheproductbasis|mol,sys(cid:105)={|00(cid:105),|01(cid:105),|10(cid:105),|11(cid:105)}
t
is given by
cosφ −sinφ
sinφ cosφ
Uexp(iφσ )= . (13)
y sinφ cosφ
cosφ −sinφ
The two Kraus matrices in this representation are simply the left block matrices of U:
(cid:18) (cid:19) (cid:18) (cid:19)
cosφ sinφ
M = M = . (14)
0 sinφ 1 cosφ
The non-selective Markov chain (1) is as trivial as [? ]
(cid:37)t+1 =(cid:37)t (cid:37)t+1 =(cid:37)t (cid:37)t+1 =sin(2φ)(cid:37)t (15)
00 00 11 11 01 01
i.e., the diagonal elements are preserved while the off-diagonals will step towards zero unless we took the singular
molecular states with φ=π/4:
(cid:18)(cid:37)0 0 (cid:19)
(cid:37) = 00 . (16)
∞ 0 (cid:37)0
11
We can say that our Markov chain with φ(cid:54)=π/4 is asymptotically equivalent with a single von Neumann projective
measurement. Evenifthesinglecollisionsareintheweakmeasurementregime,theircumulativeeffectistheprojective
measurement, as is well known, e.g., from [9–11].
B. Repeated XORs with time overlap
To construct the simplest non-Markov chain, let each molecule interact with the system qubit twice, and let there
be an overlap between subsequent collisional periods, see Fig. 5(a). We are interested in stationary chains hence the
patternofdoublecollisionwillidenticallyrepeatitselfalongthechain. Note, however, thatthestartingpatternmust
always be a “broken” one, i.e., in the beginning there is a molecule that collides only once with the central system.
From the classical theory of NM chains with finite memory time we know that adding suitable amount of memory
(bits) to the system yields an equivalent Markovian chain. Similarly, we attach a memory qubit to the system qubit.
We can make e.g., one distinguished molecule the memory. Consider the resulting scheme in Fig. 5(b), equivalent
with Fig. 5(a). The system+memory compound is a two-qubit composite system whose state will be denoted by (cid:37)˜
t
and, as we see, it undergoes independent collisions with the rest of the molecules of state ξ each. Our trick is that we
repeatedly swap all system-environment entanglement into the system-memory compound.
Accordingly, we have obtained a Markovian quantum chain for the system+memory compound (cid:37)˜ instead of the
t
system (cid:37) alone. Markovian structure (1)-(4) applies invariably. The molecule-memory-system compound is a three-
t
qubit system. The three-qubit unitary operation U is the following:
U =XOR SWAP XOR . (17)
sys−mol mol−mem sys−mol
In the basis |mol,mem,sys(cid:105) = {|000(cid:105),|001(cid:105),|010(cid:105),...|111(cid:105)}, we calculate the collision matrix (17) times exp(iφσ ),
y
with notations c=cosφ and s=sinφ:
c −s
s c
s c
c −s
Uexp(iφσ )= . (18)
y c −s
s c
s c
c −s
6
λ
ξ ξ
ρ
ξ
λ λ
(a)NMqubitchain
~{
ρ
ξ ξ ξ
λ λ λ
(b)EquivalentMarkovchain
FIG. 5. Equivalent one-qubit NM and two-qubit Markov chains. (a) NM chain: the system qubit (cid:37) interacts (collides) twice
withthesamereservoirqubits(molecules)ξ viatwoXOR-gates,withoverlapbetween’nearest’molecules;thepost-interaction
statesofthemolecularqubitsaremeasured,yieldingtherandomoutcomesλ. (b)EquivalentMarkovchain: thesystem-satellite
two-qubit(cid:37)˜compoundinteracts(collides)withthereservoirqubits(molecules)ξviatheswap-andtwoXOR-gatesbutwithout
theoverlapbetweencollisionalintervalsofdifferentmolecules;thepost-interactionstatesofthemolecularqubitsaremeasured,
yielding the random outcomes λ.
According to (12), the two Kraus matrices M and M in the basis |mem,sys(cid:105)={|00(cid:105),|01(cid:105),|10(cid:105),|11(cid:105)} are given by
0 1
the upper-left and lower-left 4×4 blocks, respectively:
c 0 0 c
0 s s 0
M = M = . (19)
0 s 0 1 0 s
0 c c 0
Using these Kraus matrices, the non-selective Markov chain (1) reads:
c2((cid:37)˜t +(cid:37)˜t ) cs((cid:37)˜t +(cid:37)˜t ) cs((cid:37)˜t +(cid:37)˜t ) c2((cid:37)˜t +(cid:37)˜t )
00 22 03 21 00 22 03 21
cs((cid:37)˜t +(cid:37)˜t ) s2((cid:37)˜t +(cid:37)˜t ) s2((cid:37)˜t +(cid:37)˜t ) cs((cid:37)˜t +(cid:37)˜t )
(cid:37)˜ = 30 12 11 33 30 12 11 33 . (20)
t+1 cs((cid:37)˜t +(cid:37)˜t ) s2((cid:37)˜t +(cid:37)˜t ) s2((cid:37)˜t +(cid:37)˜t ) cs((cid:37)˜t +(cid:37)˜t )
00 22 03 21 00 22 03 21
c2((cid:37)˜t +(cid:37)˜t ) cs((cid:37)˜t +(cid:37)˜t ) cs((cid:37)˜t +(cid:37)˜t ) c2((cid:37)˜t +(cid:37)˜t )
30 12 11 33 30 12 11 33
On the r.h.s. we can identify the diagonal part of the system density matrix (cid:37)t =((cid:37)˜t +(cid:37)˜t ) and (cid:37)t =((cid:37)˜t +(cid:37)˜t )
00 00 22 11 00 33
as well as the specific correlation Ct =Tr(σ ⊗σ (cid:37)˜)=((cid:37)˜t +(cid:37)˜t ) between the memory and the system. Observe
x− x − t 03 21
that all these quantities are invariants along our NM chain. Not surprisingly then, the chain will immediately reach
the stationary state (cid:37)˜ =(cid:37)˜ ···=(cid:37)˜ fully parametrized by the above invariants:
1 2 ∞
c2(cid:37)0 csC0 cs(cid:37)0 c2C0
00 x− 00 x−
csC0 s2(cid:37)0 s2C0 cs(cid:37)0
(cid:37)˜ = x+ 11 x+ 11. (21)
∞ cs(cid:37)0 s2C0 s2(cid:37)0 csC0
00 x− 00 x−
c2C0 cs(cid:37)0 csC0 c2(cid:37)0
x+ 11 x+ 11
The free parameter C0 depends on the starting “broken” pattern. In the case shown on Figs. 5(a) and 5(b), it
x±
can be set to zero, assuming uncorrelated initial satellite memory with (cid:104)σ (cid:105) = 0. The above stationary state of the
x
memory+system compound then becomes a separable, disentangled state:
(cid:37)˜∞ =(cid:37)000|Ψφ(cid:105)(cid:104)Ψφ|⊗|0(cid:105)(cid:104)0|+(cid:37)011(cid:12)(cid:12)Ψ(cid:48)φ(cid:11)(cid:10)Ψ(cid:48)φ(cid:12)(cid:12)⊗|1(cid:105)(cid:104)1|, (22)
7
where the memory state |Ψ(cid:48)(cid:105)=s|0(cid:105)+c|1(cid:105) is not orthogonal to |Ψ (cid:105)=c|0(cid:105)+s|1(cid:105) (unless φ=π/4). The stationary
φ φ
density matrix (cid:37) =(cid:37) ···=(cid:37) of our central system is trivial:
1 2 ∞
(cid:18)(cid:37)0 0 (cid:19)
(cid:37) =Tr (cid:37)˜ = 00 . (23)
∞ mem ∞ 0 (cid:37)0
11
It is the mixture of the basis states with the initial population, as if (cid:37) were ideally measured by the collective of the
0
molecules, like in case of the Markovian chain, cf. Sec. IVA.
Therepeated-XORmodelhastrivialinvariants,thisisthereasonitreachestheasymptoticstateinonestepalready.
C. Distributed XORs with time overlap
In order to make a NM chain evolve asymptotically towards the stationary state we must assure that there are
fewer trivial invariants. Let us go back for a moment to the Markovian chain with single XOR collisions (13) and
introduceaNMvariant. SupposethateachXOR-collisiontakesafinitetimeandsubsequentXORoperationsoverlap
in time. To construct a simplest discrete model of time-distributed XOR, we make XOR in two steps separated by
unit time. Let both steps correspond to the following square-root of XOR:
1
√ 1
XOR= (cid:112) (cid:112) . (24)
i/2 −i i/2
(cid:112) (cid:112)
−i i/2 i/2
√
(cid:112)
(Here the convention i/2 = eiπ/8/ 2 has been chosen.) Our non-Markov model is shown in Fig. 5(a), with the
√
XOR collisions replaced by the above XOR.
Obviously, we have the equivalent Markov chain, like in Fig. 5(b), and we can perform the same calculations as
before. We calculate the unitary matrix
√ √
U = XOR SWAP XOR (25)
sys−mol mol−mem sys−mol
times exp(iφσ ), and read out the two Kraus matrices:
y
c 0 0 c
iβ¯ β¯ β¯ iβ¯
M = M = , (26)
0 s 0 1 0 s
β −iβ −iβ β
withβ =eiφ/2. TheseKrausmatricesin(2)yield, afterdirectcalculations, thefollowingMarkovianchain(1)forthe
memory-system compound:
c2((cid:37)˜t +(cid:37)˜t ) cβ∆ cs((cid:37)˜t +(cid:37)˜t ) icβ¯∆
00 22 t 00 22 t
(cid:37)˜t+1=cs((cid:37)˜ct0β¯0∆+¯t(cid:37)˜t22) 1/2(s(cid:37)˜βt11∆+t(cid:37)˜t33) s2((cid:37)˜st0β¯0∆+¯t(cid:37)˜t22) 2iβ¯2i(s(cid:37)˜β¯t11∆+t(cid:37)˜t33), (27)
−cβi∆¯ −2iβ2((cid:37)˜t +(cid:37)˜t ) −sβi∆¯ 1/2((cid:37)˜t +(cid:37)˜t )
t 11 33 t 11 33
where
∆ =−i((cid:37)˜t +(cid:37)˜t )+((cid:37)˜t +(cid:37)˜t ) (28)
t 01 23 03 21
and it satisfies a closed evolution equation:
∆ =sin(2φ)∆ , (29)
t+1 t
therefore this is a convenient parametrization. It follows from (27) that the matrix elements not containing ∆ will
taketheirfinalstationaryvaluesimmediatelyafterthefirststep,justlikeinourpreviousmodelinSec.IVB.However,
the elements with ∆ show an exponential relaxation (29) toward zero if sin(2φ) (cid:54)= 0. The relaxation of ∆ governs
t
the asymptotic diagonalization of the system density matrix:
(cid:18) (cid:37)t=0 (1+isin(2φ))∆ /2(cid:19)
(cid:37) =Tr (cid:37)˜= 00 t . (30)
t mem (1−isin(2φ))∆¯ /2 (cid:37)t=0
t 11
8
Indeed, for t = ∞ one has ∆ = 0, the system density matrix (cid:37) becomes diagonal with the initial populations
∞ ∞
(cid:37)0 ,(cid:37)0 . As to the stationary density matrix (cid:37)˜ of the memory-system compound, the Markov chain (27) yields a
00 11 ∞
separable state again, like in our previous model in Sec. IVB. This time we get
(cid:37)˜∞ =(cid:37)000|Ψφ(cid:105)(cid:104)Ψφ|⊗|0(cid:105)(cid:104)0|+(cid:37)011(cid:12)(cid:12)Ψ(cid:48)φ(cid:11)(cid:10)Ψ(cid:48)φ(cid:12)(cid:12)⊗|1(cid:105)(cid:104)1|, (31)
√
wherethememorystate|Ψ(cid:48)(cid:105)=(|0(cid:105)+ie2iφ|1(cid:105))/ 2,differentfrom|Ψ(cid:48)(cid:105)inthemodelinSec.IVB,isneverorthogonal
φ φ
to |Ψ (cid:105). This is what makes our system quantum NM, a distinction discussed in the next section.
φ
V. CLASSICAL/QUANTUM NON-MARKOVIANITY
Our construction of NM quantum chains may become universal if we allow for more than two repeated collisions
with a single molecule and/or overlaps of collision periods between more than two molecules. Whether or not an
arbitrary NM chain will be reducible to ours remains an open theoretical issue. Our class of NM chains is unique for
at least one thing: we can always identify the satellite memory to make the time evolution of the resulting system-
memory compound a Markovian chain. We can always read-out from the circuit of the NM chain how many qubits
areneededforthesatellitememory. Thisnumberofqubitsisausefulcount ofnon-Markovianityofourchain, e.g., in
computationalsimulationthisnumbergivesanupperboundonhowmuchdatashouldbestoreddynamicallytogether
with the data of our system of interest.
However, this count may be significantly larger than the amount of information to be contained in the satellite.
Suppose that we have determined the minimum number of qubits needed for the satellite, so the count of non-
Markovianityisknown. Thenasuitableinformaticquantity,likethemutualinformation[5,6]ofthesatellitememory
(M) and the system (S) might play the role of non-Markovianity measure µ . In the stationary regime one gets
NM
µ =I(S :M)=H(S)+H(M)−H(S,M) (32)
NM
where H stands in turn for the von Neumann entropy of the S-state Tr (cid:37)˜ =(cid:37) , of the M-state Tr (cid:37)˜ , and of
mem ∞ ∞ sys ∞
the composite state (cid:37)˜ .
∞
We should stop for a second, and distinguish quantum from classical non-Markovianity. The attentive reader may
havenoticedthatourconstructionofquantumNMchainsinitselfhasnothingparticularforquantumchains,itroots
in a similar construction of classical NM chains. In fact, any classical NM chain can be represented by an equivalent
Markovian chain if we assign a sufficient satellite memory. The minimum size of the satellite memory (e.g.: in bits) is
the natural count of non-Markovianity, the mutual information may be the measure of non-Markovianity. In the case
of a generic quantum NM chain, it may happen that the minimum satellite memory still consists of bits, qubits are
not required at all. In this case we say that the chain is classically NM, and its quantum non-Markovianity is zero.
It is remarkable that both NM chains in Sec. IV have turned out to be quantum NM. If we look at the composite
states (22,31) of the system+satellite compound, we see that classical satellite bits would not work, we need satellite
qubits. Although we got zero stationary entanglement between the system and satellite, it does not mean the lack of
quantumness. The price of getting rid of quantum non-Markovianity cannot be paid in classical bit instead of qubit.
The count of quantum non-Markovianity is 1 for both NM chains. In order to distinguish quantum from classical
non-Markovianity we use the notion of discord [21]. The classical measure of non-Markovianity may be defined by
µcl =J(S :M) =H(S)−H(S|{ΠM}) (33)
NM {ΠM} j
j
where the rightmost term means the average von Neumann entropy of S when the satellite memory is undergoing the
projective measurement via the set ΠM. The quantum non-Markovianity measure is the discord itself:
j
µqu =I(S :M)−J(S :M) . (34)
NM {ΠM}
j
The sum of quantum and classical non-Markovianity measures yields the total measure µ .
NM
(cid:12) (cid:69)
We can check our proposal on the two models of Sec. IV. Since the memory states |Ψ (cid:105),(cid:12)Ψ(cid:48) are non-orthogonal,
φ (cid:12) φ
we get non-zero discord. If they were orthogonal, we could get zero discord, and also we could measure the memory
after each collision so that a single classical bit 0,1 could be retained instead of the memory qubit: the chain would
be classically NM in the stationary regime, with non-Markovianity count 1 (bit).
9
ρ
ξ ξ ξ ξ
FIG. 6. NM quantum chain of advanced overlap: the system interacts with each independent molecule twice, with overlap
between collisional intervals of ’nearest’ and ’next-to-nearest’ molecules. (Boxes connected with vertical lines represent the
bipartite unitary collisions.)
VI. BRIEF SUMMARY AND OUTLOOK
In the framework of the abstract collision model of system-reservoir interactions, we constructed quantum chains
meeting the minimal requirements for NM stationary time evolution. A method of systematic construction of the
equivalent Markovian quantum chains using explicit memory allocation is discussed. Due to the transparent NM
structure, we can always identify a well-defined part of the reservoir as the memory, this part is called the satellite
memory of the system. The time evolution of the system-plus-satellite-memory compound is Markovian. We suggest
a novel distinction: a given quantum NM chain is either quantum NM (if the satellite requires qubits) or classical
NM (if bits suffice for the satellite). Accordingly, we suggest the numbers of satellite qubits and bits, respectively,
as counts of quantum/classical non-Markovianity of a given quantum chain. The mutual information and discord is
proposed to measure non-Markovianity and quantum non-Markovianity, respectively.
Two examples of one-qubit NM chains are discussed. In both examples, the corresponding satellite memory is a
single qubit. The stationary state is exactly calculable, the non-vanishing discord of the satellite qubit w.r.t. the
system qubit indicates that our examples are quantum non-Markovian.
Although our calculations are restricted for non-selective NM chains, the structural transparency allows for the
selective refinements. Option of monitoring the NM system is inherent in the model.
For achieving a higher level non-Markovianity than discussed in our work (Fig. 1(b)), one can construct a similar
chain with more collisions per molecule and/or longer intervals of overlap. In Fig. 6, e.g., the number of collisions
is invariably two whereas the lengths of overlapping intervals have been increased for three collisions. In return, the
equivalentMarkovianstructureneedstwosatellitequbitsinsteadofone,thenon-Markovianitycountis2. Theability
to handle complex NM processes may pave the way to the construction of environmental interaction models which
better approximate reality.
Support by the Hungarian Scientific Research Fund under Grant No. 75129 and support by the EU COST Action
MP100 are acknowledged. The authors thank Tam´as Geszti for useful discussions.
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